## The Annals of Probability

### Large Deviations, Moderate Deviations and LIL for Empirical Processes

Liming Wu

#### Abstract

Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d. r.v.'s with values in a measurable space $(E, \mathscr{E})$ of law $\mu$, and consider the empirical process $L_n(f) = (1/n)\sum^n_{k=1} f(X_k)$ with $f$ varying in a class of bounded functions $\mathscr{F}$. Using a recent isoperimetric inequality of Talagrand, we obtain the necessary and sufficient conditions for the large deviation estimations, the moderate deviation estimations and the LIL of $L_n(\cdot)$ in the Banach space of bounded functionals $\mathscr{l}_\infty(\mathscr{F})$. The extension to the unbounded functionals is also discussed.

#### Article information

Source
Ann. Probab., Volume 22, Number 1 (1994), 17-27.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176988846

Digital Object Identifier
doi:10.1214/aop/1176988846

Mathematical Reviews number (MathSciNet)
MR1258864

Zentralblatt MATH identifier
0793.60032

JSTOR